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Linear Operators - Basis and Dimension - year 4

Question (1)
Find a basis and dimension of the subspace W of R4 generated by the vectors
( 1 , - 4 , - 2 , 1 ) , ( 1 , - 3 , - 1 , 2 ) , ( 3 , - 8 , - 2 , 7 ) .
Extend it to find the basis of R4 .

Question (2)
Determine a basis and the dimensions of the Subspace of M2(R) generated by the
2 by 2 Matrices [ 2 -10 ] , [ 3 3 ] , [ 2 -4 ] , [ 2 -14 ]
[ -8 4 ] [ -3 15] [ -5 7] [ -10 2 ]

NOTE : For the full description of the question in mathematical font and format, please download the attached question file.

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Solution (1)
Reducing the given matrix to Echelon form in the following way , we get the matrix ..............
Hence the dimension of W = 2

The basis of W = { ( 1 , - 4 , - 2 , 1 ), ( 0 , 1 , 1 , 1 ) }

The basis of R4 = ...

Solution Summary

Solutions to the posted problems are given with step by step instructions so that the students could easily understand the solutions and use these solutions to solve other similar problems. The problems are from Linear Algebra. For complete description of the problems, please see the problems. Solutions are given in the attached file.

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